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```CHAP, vi]                    PERIODIC DECIMAL FRACTIONS.                          169
Karl Broda68 considered a periodic decimal fraction F having an even number r of digits in the period and a number w of p digits preceding the period. Let x be the first half of the period, y the second half. Then
JLj __ 5 __ I     y     i   ' x     i       = P   I m       m+r       m+2r"rw+3r    ' * '       m
__ __                                  =
10m    10m+r    10m+2r"r10w+3r    ' * '    10m    10m(102r~l)
_9(p-l(f+x+p)+a
— l)/9 = a. . .a (to r terms).   The first paper treated the case 0, and gave the generalization to base a in place of 10:
x . y  . x               a+(a — l)s      ,r    .         ar-l
-
The case a=a — 1 shows that a purely periodic fraction to the base a equals (z+l)/(ar+l) if the sum of the half periods has all its digits (to base a) equal to a— 1. Returning to the base 10, and taking N= 9(10r+l), Z = 9z+a, where each digit of 3 is ^a, we see that Z/N equals a decimal fraction in which x is the first half of the period of r digits, while the second hah" is such that the sum of corresponding digits in it and x is a. If R is the remainder after r digits of the period have been obtained, R+Z=a (10r+l)«
C.  G. Reuschle69 gave tables which serve to find numbers belonging to a given exponent < 100 with respect to a given prime modulus < 1000.
P. Mansion70 gave a detailed proof that, if n is prime to 2, 3, 5, and if the period for 1/n has n— 1 digits, the sum of corresponding digits in the half periods is 9.
T. Muir71 proved that, If p is a prime, either of
Nx= 1 (mod p1),          2V*"n=l (mod p°+n)
follows from the other. If Xi is the least positive integer x for which the first holds and if p8 is the highest power of p dividing NXl — 1, then Xipn is the least positive integer y for which Ny= 1 (mod p*+n). Hence the known theorem: If N=Hpini} where pi, p2, . . . are distinct primes, and if the period for I/ pi has mt digits, and if p* is the highest power of Pi dividing 10™* — 1, the number of digits in the period for l/N is the 1. c. m. of the mip"^*. He asked if 6 = 1 when p>3, as affirmed by Shanks.65.
Mansion's proof (ibid., 5, 1876, 33) by use of periodic decimals of the generalized Fermat theorem is quoted under that topic.
D. M. Sensenig72 noted that a prime p5^2, 5, divides N if it divides the sum of the digits of N taken in sets of as many figures each as there are digits in the period for I/ p.
"Archiv Math. Phys., 56, 1874, 85-£8; 57, 1875, 297-301.
"Tafeln complexer Primzahlen, Berlin, 1875.   Errata by Cunningham, Mess. Math., 46,
1916, 60-1.
70Nouv. Corresp. Math., 1, 1874-5, 8-12. "Messenger Math.. 4, 1875, 1-5. "The Analyst, Des Moines, Iowa, 3, 1876, 25.```